Error Bounds for Lanczos Approximations of Rational Functions of Matrices
Numerical Validation in Current Hardware Architectures
On the use of matrix functions for fractional partial differential equations
Mathematics and Computers in Simulation
Shift-Invert Arnoldi Approximation to the Toeplitz Matrix Exponential
SIAM Journal on Scientific Computing
On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems
SIAM Journal on Scientific Computing
On the Convergence of Krylov Subspace Methods for Matrix Mittag-Leffler Functions
SIAM Journal on Numerical Analysis
Using the Restricted-denominator Rational Arnoldi Method for Exponential Integrators
SIAM Journal on Matrix Analysis and Applications
Computation of matrix functions with deflated restarting
Journal of Computational and Applied Mathematics
A family of Adams exponential integrators for fractional linear systems
Computers & Mathematics with Applications
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In this paper we investigate some well-established and more recent methods that aim at approximating the vector $\exp(A)v$ when $A$ is a large symmetric negative semidefinite matrix, by efficiently combining subspace projections and spectral transformations. We show that some recently developed acceleration procedures may be restated as preconditioning techniques for the partial fraction expansion form of an approximating rational function. These new results allow us to devise a priori strategies to select the associated acceleration parameters; theoretical and numerical results are shown to justify these choices. Moreover, we provide a performance evaluation among several numerical approaches to approximate the action of the exponential of large matrices. Our numerical experiments provide a new, and in some cases, unexpected picture of the actual behavior of the discussed methods.